Characterization of the Hardy–Littlewood–Pólya principle for r.i. quasi-Banach function spaces

Characterize the rearrangement-invariant quasi-Banach function norms over resonant measure spaces for which the Hardy–Littlewood–Pólya principle holds, i.e., establish necessary and sufficient conditions ensuring that f ≺ g implies ||f|| ≤ ||g|| for all measurable functions f and g. The goal is to provide a complete characterization analogous to the classical Banach function space case.

Background

The paper reviews the Hardy–Littlewood–Pólya (HLP) relation f ≺ g (defined by f^{**} ≤ g^{**}) and the associated HLP principle (monotonicity of an r.i. quasinorm under ≺). It recalls that the HLP principle is classical for r.i. Banach function norms but notes the absence of a complete theory in the quasi-Banach setting.

The authors develop representation tools (canonical representation quasinorm) and several characterizations relating properties of X and its representation \overline{X}, including the inheritance of properties under rearrangement and majorization, but point out that a full characterization of when HLP holds for r.i. quasi-Banach function spaces is missing.

This open problem is central to extending monotonicity principles and interpolation mechanisms beyond the Banach framework, and would clarify when results relying on HLP (e.g., maximal inequalities or ergodic limits) can be transferred to quasi-Banach contexts.